How is the Nambu Goto action proportional to the world sheet area?

(I'm not sure if this belongs in the BSM forum...apologies to the moderator if it belongs there.)

I'm working through Polchinski's book on string theory (volume 1) and I came cross the definition of the Nambu Goto action. I want to understand why the Nambu Goto action is proportional to the area of the world sheet. This is probably a trivial question but I'll ask anyway.

Note the Nambu Goto action has an obvious geometric meaning but it is not the action that ends up getting quantized usually...The Polyakov action is introduced instead which is classically equivalent. Interestingly see:

maverick: I strongly suggest you to use even the Zwiebach book while studying Polchinski. It's very clear and well written, even though it doesn't use the full mathematical formalism of Polchinski.

Thank you samalkhaiat and Rexcirus.

Why does Polchinski refer to [itex]h_{ab}[/itex] as the metric (and explicitly say that it isn't the induced metric) when Zwiebach says it is actually the induced metric?

To be perfectly clear, Polchinski uses [itex]h_{ab}[/itex] in equation 1.2.9b in defining the Nambu-Goto action, whereas the notation used by Zwiebach is [itex]\gamma_{\alpha\beta}[/itex] in equation 6.44. Zwiebach calls his [itex]\gamma_{ab}[/itex] an induced metric on the world-sheet (just above equaton 6.42), but Polchinski says his [itex]h_{ab}[/itex] is a metric (as opposed to an induced metric).

Why does Polchinski refer to [itex]h_{ab}[/itex] as the metric (and explicitly say that it isn't the induced metric) when Zwiebach says it is actually the induced metric?

To be perfectly clear, Polchinski uses [itex]h_{ab}[/itex] in equation 1.2.9b in defining the Nambu-Goto action, whereas the notation used by Zwiebach is [itex]\gamma_{\alpha\beta}[/itex] in equation 6.44. Zwiebach calls his [itex]\gamma_{ab}[/itex] an induced metric on the world-sheet (just above equaton 6.42), but Polchinski says his [itex]h_{ab}[/itex] is a metric (as opposed to an induced metric).